Efficient Implementation of CFS-CPML in FDTD Solutions of Second-Order Seismic Wave Equations

The complex-frequency-shifted convolutional perfectly matched layer (CFS-CPML) has been widely used in numerical simulations of the first-order seismic wave equations to avoid artifical reflections caused by truncated boundaries. However, numerically solving the second-order seismic wave equation is preferred for some large-scale geophysical algorithms, such as reverse-time migration (RTM). Extension of CFS-CPML to the second-order seismic wave equations has been realized by using a strict derivation approach in many literatures. However, the derived CFS-CPML formulations are complex, due to introduction of a large number of intermediate variables, which decreases overall computational efficiency, visibly. We present an efficient strategy to implement CFS-CPML in finite-difference time-domain (FDTD) modeling of the second-order seismic wave equations. We do not derive any continuous CFS-CPML formulations. Instead, we split the high-order centered-grid finite-difference (CGFD) stencil applied to approximate the second-order derivatives into two-level CGFD stencils. The inner stencils are viewed as the second-order CGFD approximations of the first-order derivatives at symmetric grid points. With this viewpoint, the convolution term related to CFS-CPML can be introduced to each inner CGFD approximation, naturally. The outer CGFD stencil is also viewed as CGFD approximation of a first-order derivative and it is augmented by a convolution term as well. By this way, CFS-CPML is incorporated into the FDTD simulations, efficiently. We take three-dimensional (3D) acoustic, viscoacoustic and elastic wavefields modeling for examples to verify the feasibility of the new implementation strategy of CFS-CPML.